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Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. (English) Zbl 0924.35051
The author deals with a fully nonlinear second-order, possibly degenerate, elliptic equation (1) $$F(x,u,Du,D^2u)=0$$ in $$\Omega$$ in a smooth bounded domain $$\Omega\subset \mathbb{R}^N$$ with fully nonlinear Neumann boundary conditions (2) $$L(x,u,Du)=0$$ on $$\partial\Omega$$. Here $$F$$ and $$L$$ are real continuous functions of all the variables $$x\in\overline\Omega$$, $$u\subset\mathbb{R}$$, $$Du$$ and $$D^2u$$ are the gradient and the Hessian matrix, respectively. The author proves comparison results between viscosity sub- and supersolutions of the elliptic problem (1)–(2) as well as of the parabolic equation $$u_t+ F(x,t,u, Du,D^2u)=0$$ in $$\Omega\times (0,T)$$ with the initial condition $$u(x,0)=u_0(x)$$ on $$\overline\Omega$$ and the boundary condition (2) or $$u_t+L (x,t,u,Du)=0$$ on $$\partial\Omega \times(0,T)$$. The comparison results have been obtained both for the case of a standard and a degenerate equation. Later on, they are applied by proving the existence and uniqueness of solutions for boundary value problems and extending the so-called level set approach for equations set in bounded domains with nonlinear Neumann boundary conditions.

##### MSC:
 35J70 Degenerate elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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##### References:
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